R E S E A R C H
Open Access
On impulsive partial differential equations
with Caputo-Hadamard fractional derivatives
Xianmin Zhang
**Correspondence: [email protected]
School of Electronic Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
Abstract
In this paper, the mixed Caputo-Hadamard fractional derivative is introduced based on the Caputo-type modification of Hadamard fractional derivatives in the existing paper, and impulsive partial differential equations with Caputo-Hadamard fractional derivatives are studied. The formula of a general solution for these impulsive fractional partial differential equations is found by considering some limiting cases (impulses tending to zero), and its validity is shown by an example.
MSC: 34A08; 34A37
Keywords: impulsive fractional partial differential equations; fractional partial differential equations; impulse; general solution
1 Introduction
The fractional calculus was developed within the frame of the Hadamard fractional deriva-tive in [–], and for the general theory of Hadamard fractional calculus we refer the in-terested reader to []. Moreover, some progress was achieved in controllability, some new definitions, some new methods of numerical solution etc. for fractional differential equa-tions [–].
Recently, Jaradet al.presented the definition of Caputo-Hadamard fractional deriva-tive in [], and developed the fundamental theorem of fractional calculus in the Caputo-Hadamard setting in [, ].
Furthermore, Vityuk and Golushkov were concerned with the existence and uniqueness of solution for a kind of fractional partial differential equations in []. Next, Abbas and Benchohra first considered fractional partial differential equations with impulses in [], and the authors gave some results as regards the existence and uniqueness of solution for these impulsive systems in [–].
Now the equivalent integral equations were found for several fractional-order systems with impulses in [–], and the obtained results show that there is a general solution for their impulsive fractional-order systems.
Motivated by the above-mentioned work, we will give the definition of a mixed Caputo-Hadamard fractional derivative and seek the equivalent integral equations for a kind of impulsive partial differential equations with Caputo-Hadamard fractional derivatives to find the essential result that there exists a general solution for impulsive fractional
ential equations in this paper. We have
⎧ ⎪ ⎨ ⎪ ⎩
(C-HDq(a+,b+)u)(x,y) =f(x,y,u(x,y)), (x,y)∈Jandx=xi(i= , , . . . ,m),
u(x+
i,y) =u(x–i,y) +Ii(u(x–i,y)), i= , , . . . ,m,
u(x,b) =φ(x), u(a,y) =ψ(y), x∈[a,A],y∈[b,B],
()
whereJ= [a,A]×[b,B] (a,b> ),q= (q,q) (hereq,q∈Cand ((q),(q))∈(, ]×
(, ]),C-HDq(a+,b+)denotes the Caputo-Hadamard fractional derivative of orderq. We have
the impulsive pointsa=x<x<· · ·<xm<xm+=A. u(x+i,y) =limε→+u(xi+ε,y) and u(x–
i,y) =limε→–u(xi+ε,y) represent the right and left limits of u(x,y) atx=xi (i=
, , . . . ,m), respectively.f :J×Cn→CnandIi:Cn→Cn(i= , , . . . ,m) are given
func-tions.φ: [a,A]→Cn,ψ: [b,B]→Cnare given continuous functions withφ(a) =ψ(b).
Consider a limiting case in system ():
lim
Ii(u(x–i,y))→ for alli∈{,,...,m}
{system ()}
→
(C-HDq(a+,b+)u)(x,y) =f(x,y,u(x,y)), (x,y)∈J,
u(x,b) =φ(x), u(a,y) =ψ(y), x∈[a,A],y∈[b,B]. ()
Therefore,
lim
Ii(u(x–i,y))→ for alli∈{,,...,m}
{the solution of system ()}
={the solution of system ()}. ()
Next, some preliminaries are given in Section , and the equivalent integral equation will be provided for a fractional partial differential system with impulses in Section . Finally, an example is presented to illuminate the main result in Section .
2 Preliminaries
In this section, we shall present the definition of Caputo-Hadamard fractional partial derivatives according to definition of left-sided Caputo-Hadamard fractional derivatives suggested by Jaradet al.in [, ], and we draw a conclusion.
Definition . Leta∈[a,A],z+= (a+,b+),Jz= [a,A]×[b,B],q= (q,q) (hereq,q∈
Cand ((q),(q))∈(, ]×(, ]). For the functionw, the expression
HJzq+w
(x,y) = (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
w(s,t)dt
t ds
s ,
whereis the gamma function, is called the left-sided mixed Hadamard integral of or-derq.
Definition . Letq= (q,q) (hereq,q∈Cand ((q),(q))∈(, ]×(, ]). Forw∈
the expression
C-HDqz+w
(x,y)
= ( –q)( –q)
x
a y
b
lnx
s
–q
lny
t –q
δsδtw(s,t) dt
t ds
s
=HJz–+qδxδyw
(x,y),
where we have the partial differential operatorδx=x∂∂x.
Lemma . Let h∈C(Jz,Cn), q= (q
,q) (here q,q∈Cand((q),(q))∈(, ]×
(, ]).A function u∈C(Jz,Cn)is a solution of the differential equation
C-HDqz+u
(x,y) =h(x,y), (x,y)∈Jz, ()
if and only if
u(x,y) =u(x,b) +ua+,y–ua+,b+Izq+h
(x,y)
=u(x,b) +ua+,y–ua+,b
+ (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
h(s,t)dt
t ds
s ,
for(x,y)∈Jz. ()
Proof Let u(x,y) is a solution of the equation (C-HDqz+u)(x,y) =h(x,y), (x,y)∈Jz. Due
to
C-HDqz+u
(x,y) =HJz–+qδxδyu
(x,y)
we have
HJzq+
HJz–+qδxδyu
(x,y) =HJzq+h
(x,y), (x,y)∈Jz.
On the other hand,
HJzq+
HJz–+qδxδyu
(x,y)
=HJz+(δxδyu)(x,y)
=u(x,y) –u(x,b) –ua+,y+ua+,b, for (x,y)∈Jz.
Therefore,
u(x,y) =u(x,b) +ua+,y–ua+,b+HJzq+h
(x,y), for (x,y)∈Jz.
3 Main results
For convenience, leti=zi= ,(x,y) =φ(x) +ψ(y) –φ(a), andf=f(s,t,u(s,t)). Define ¯
u(x,y) =u(x,b) +ux+k,y–ux+k,b
+ (q)(q)
x
xk
y
b
lnx
s
q–
lny
t q–
fdt t
ds s ,
for (x,y)∈(xk,xk+]×[b,B], andk∈ {, , . . . ,m}, ()
withu(x+
k,y) =u(x–k,y) +Ik(u(x–k,y)).
By Lemma ., it is sure thatu¯(x,y) satisfies the fractional derivative condition and im-pulsive conditions in system (). Butu¯(x,y) is not a solution of () becauseit does not satisfy
(). Therefore,u¯(x,y) will be considered an approximate solution to seek the exact solution of system ().
Theorem . Let q= (q,q),here q,q∈Cand((q),(q))∈(, ]×(, ].Ii(u(x–i,y))
(i= , , . . . ,m)are differentiable functions on y.System()is equivalent to the integral equation
u(x,y) =(x,y) + (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+
k
i=
Iiux–i,y–Iiux–i,b
+ σ(y) (q)(q)
k
i=
Ii
ux–i,y xi
a y
b
lnxi
s
q–
lny
t q–
fdt t
ds s
+
x
xi
y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
,
for(x,y)∈(xk,xk+]×[b,B]
here k∈ {, , , . . . ,m}, ()
provided that the integral in()exists,whereσ(y)is an arbitrary differentiable function on y.
Proof As regards necessity; lettingIi(u(x–
i,y))→ for alli∈ {, , . . . ,m}in equation (),
we obtain
lim
Ii(u(x–i,y))→ for alli∈{,,...,m}
u(x,y)
=(x,y) + (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s ,
for (x,y)∈(xk,xk+]×[b,B],k∈ {, , , . . . ,m}.
Therefore, by Lemma ., equation () (under conditions Ii(u(x–
i,y))→ for all i ∈
Next, for∀xi(i∈ {, , . . . ,m}) in equation (), we get
ux+i,y–ux–i,y=x+i,y+Iiux–i,y–Iiux–i,b–x–i,y
=Iiux–i,y–Iiuxi–,b+φxi+–φx–i
=Iiux–i,y.
Therefore, equation () satisfies the impulsive conditions in system ().
Finally, taking fractional derivatives of both sides of equation () as (x,y)∈(xk,xk+]×
[b,B] (herek= , , , . . . ,m), we obtain
C-HDq(a+,b+)u
(x,y)
=HJ(–a+,qb+)δxδyu
(x,y)
=HJ(–a+,qb+)δxδy
(x,y) +
k
i=
Iiux–i,y–Iiux–i,
+ (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+ σ(y) (q)(q)
k
i=
Iiux–i,y xi
a y
b
lnxi
s
q–
lny
t q–
fdt t
ds s
+
x
xi
y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
=
fx,y,u(x,y)|(x,y)∈[a,xk+]×[b,B]+
(q)(q)
×
k
i=
HJ(–a+,qb+)δxδy x
xi
lnx
s
q– y
b
σ(y)Iiux–i,ylny
t q–
fdt t
ds
s
–
x
a
lnx
s
q– y
b
σ(y)Iiux–i,ylny
t q–
fdt t
ds
s
(x,y)∈(xk,xk+]×[b,B]
.
We have
HJ(–a+,qb+)δxδy x
xi
lnx
s
q– y
b
σ(y)Iiux–i,ylny
t q–
fdt t
ds
s
–
x
a
lnx
s
q– y
b
σ(y)Ii
ux–i,ylny
t q–
fdt t
ds
s
= . ()
Also, we will give the proof of equation () in the Appendix. Thus
C-HDq(a+,b+)u
(x,y) =fx,y,u(x,y)|(x,y)∈(xk,xk+]×[b,B].
As regards sufficiency: we will prove that the solution of system () satisfies equation () by mathematical induction. By Lemma ., the solution of system () satisfies
u(x,y) =(x,y) + (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s ,
for (x,y)∈[a,x]×[b,B]. ()
Using (), the approximate solution (as (x,y)∈(x,x]×[b,B]) of system () is given by
¯
u(x,y) =u(x,b) +ux+,y–ux+,b
+ (q)(q)
x
x y
b
lnx
s
q–
lny
t q–
fdt t
ds s
=φ(x) +φx–+ψ(y) –φ(a) +I
ux–,y
+ (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–φx––ψ(b) +φ(a) –I
ux–,b
+ (q)(q)
x
x y
b
lnx
s
q–
lny
t q–
fdt t
ds s
=(x,y) +I
ux–,y–I
ux–,b
+ (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+
x
x y
b
lnx
s
q–
lny
t q–
fdt t
ds s
,
for (x,y)∈(x,x]×[b,B]. ()
Lete(x,y) =u(x,y) –u¯(x,y) for (x,y)∈(x,x]×[b,B], hereu(x,y) denotes the exact
so-lution of system (). Moreover, by equation (), the exact soso-lutionu(x,y) of system () satisfies
lim
I(u(x–,y))→u(x,y) =(x,y) +
(q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s ,
for (x,y)∈(x,x]×[b,B]. ()
Thus,
lim
I(u(x–,y))→ e(x,y)
= lim
I(u(x–,y))→
u(x,y) –u¯(x,y)
= (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
x y
b
lnx
s
q–
lny
t q–
fdt t
ds s
Equation () means thate(x,y) is connected withlimI(u(x–,y))→e(x,y) andI(u(x–,y)).
Therefore, we suppose
e(x,y) =κ
whereκis an undetermined function withκ() = . Thus,
u(x,y) =u¯(x,y) +e(x,y)
() is provided by
Lete(x,y) =u(x,y) –u¯(x,y) for (x,y)∈(x,x]×[b,B]. Moreover, by equation (), the
exact solution of () satisfies
+ –κ(I(u(x
– ,y)))
(q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+
x
x y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+ –κ(I(u(x
– ,y)))
(q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+
x
x y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
,
for (x,y)∈(x,x]×[b,B]. ()
On the other hand, for system (), we have
lim
x→x ⎧ ⎪ ⎨ ⎪ ⎩
(C-HDq(a+,b+)u)(x,y) =f(x,y,u(x,y)), (x,y)∈Jandx=x,x,
u(x+i,y) =u(x–i,y) +Ii(u(x–i,y)), i= , ,
u(x,b) =φ(x), u(a,y) =ψ(y), x∈[a,A],y∈[b,B]
()
=
⎧ ⎪ ⎨ ⎪ ⎩
(C-HDq(a+,b+)u)(x,y) =f(x,y,u(x,y)), (x,y)∈Jandx=x,
u(x+,y) =u(x–,y) +I(u(x–,y)) +I(u(x–,y)),
u(x,b) =φ(x), u(a,y) =ψ(y), x∈[a,A],y∈[b,B].
()
Using () and () to () and (), respectively, we get
–κI
ux–,y+I
ux–,y= –κI
ux–,y+ –κI
ux–,y, for∀I
ux–,yandI
ux–,y. ()
Therefore, –κ(Ii(u(x–i,y))) =σ(y)Ii(u(x–i,y)), hereσ(y) is a differentiable function ony.
Thus, () and () can be rewritten into
u(x,y) =(x,y) +I
ux–,y–I
ux–,b
+ (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+σ(y)I(u(x
– ,y))
(q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+
x
x y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
,
for (x,y)∈(x,x]×[b,B], ()
u(x,y) =(x,y) +I
ux–,y–I
ux–,b+I
ux–,y–I
+ σ(y) (q)(q)
n
i=
Iiux–i,y xi
a y
b
lnxi
s
q–
lny
t q–
fdt t
ds s
+
xn+
xi
y
b
lnxn+
s
q–
lny
t q–
fdt t
ds s
–
xn+
a y
b
lnxn+
s
q–
lny
t q–
fdt t
ds s
,
for (x,y)∈(xn+,xn+]×[b,B]. ()
Leten+(x,y) =u(x,y) –u¯(x,y) for (x,y)∈(xn+,xn+]×[b,B], hereu(x,y) denotes the exact
solution of system (). Moreover, by equation (), the exact solution satisfies
lim
Ii(u(x–i,y))→,
for alli∈{,,...,n+}
u(x,y) =(x,y) + (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s ,
for (x,y)∈(xn+,xn+]×[b,B], ()
lim
Ij(u(x–j,y))→,
herej∈{,,...,n+}
u(x,y)
=(x,y) +
≤i≤n+, andi=j
Iiux–i,y–Iiux–i,b
+ (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+ σ(y) (q)(q)
≤i≤n+, andi=j
Iiux–i,y xi
a y
b
lnxi
s
q–
lny
t q–
fdt t
ds s
+
x
xi
y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
,
for (x,y)∈(xn+,xn+]×[b,B]. ()
Thus,
lim
Ii(u(x–i,y))→,
for alli∈{,,...,n+}
en+(x,y)
= lim
Ii(u(x–i,y))→,
for alli∈{,,...,n+}
u(x,y) –u¯(x,y)
= (q)(q)
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
xn+
a y
b
lnxn+
s
q–
lny
t q–
fdt t
ds s
–
x
xn+ y
b
lnx
s
q–
lny
t q–
fdt t
ds s
+ σ(y) (q)(q)
n+
i=
Iiux–i,y xi
a y
b
lnxi
s
q–
lny
t q–
fdt t
ds s
+
x
xi
y
b
lnx
s
q–
lny
t q–
fdt t
ds s
–
x
a y
b
lnx
s
q–
lny
t q–
fdt t
ds s
,
for (x,y)∈(xn+,xn+]×[b,B]. ()
Therefore, the solution of system () satisfies equation (). Thus, by necessity and suffi-ciency, system () is equivalent to equation (). The proof is completed.
4 Examples
In this section, we will give an example to reveal that there exists a general solution for impulsive fractional partial differential equations.
Example . Let us consider the impulsive fractional system
⎧ ⎪ ⎨ ⎪ ⎩
(C-HDq(+,+)u)(x,y) =lnxlny, (x,y)∈[, ]×[, ] andx=
u(+,y) =u(–,y) +ly,
u(x, ) =u(,y)≡, x∈[, ],y∈[, ],
()
whereq= (+j,+j) (herejdenotes the imaginary unit) andlis a constant. By Theo-rem ., the general solution of () is given by
u(x,y) = (+j)(+j)
x
y
lnx
s
+j–
lny
t
+j–
lntlnsdt t
ds s
=
(+j)(+j)(lnx)
+j(lny)+j, for (x,y)∈(, ]×(, ], ()
u(x,y) =ly+ (
+j)( +j)
x
y
lnx
s
+j–
lny
t
+j–
lntlnsdt t
ds s
+ σ(y)ly (
+j)( +j)
y
ln
s
+j–
lny
t
+j–
lntlnsdt t
ds s
+
x
y
lnx
s
+j–
lny
t
+j–
lntlnsdt t
ds s
–
x
y
lnx
s
+j–
lny
t
+j–
lntlnsdt t
ds s
=ly+
(+j)(+j)(lnx)
+j
x>
(lny)+j y>
+ σ(y)ly (+j)(+j)
(ln)+j+
lnx+
+j
ln
lnx
+j
x>
– (lnx)+j x>
(lny)+j y>
whereσ(y) is a differentiable function onyin equation (). Next, we will verify that the general solution ()-() satisfies all conditions in system (). By the Appendix, we have
(i) C-HDq(+,+)
Taking fractional derivatives of the two sides of equations ()-(), we have
(using (i))
=lnxlny, for (x,y)∈(, ]×(, ].
Therefore, equations ()-() satisfy the fractional derivative condition in system (). Next, by equations ()-(), we have
(using () and ())
=ly|y∈(,].
Therefore, equations ()-() satisfy the impulsive conditions in system (). Finally, for system (), we have
lim
l→
⎧ ⎪ ⎨ ⎪ ⎩
(C-HDq(+,+)u)(x,y) =lnxlny, (x,y)∈[, ]×[, ] andx= ,
u(+,y) =u(–,y) +ly,
u(x, ) =u(,y)≡, x∈[, ],y∈[, ]
=
(C-HDq(+,+)u)(x,y) =lnxlny, (x,y)∈[, ]×[, ],
u(x, ) =u(,y)≡, x∈[, ],y∈[, ]. ()
On the other hand, by equations ()-(), we have
lim
l→{equations ()-()}
⇒ lim
l→u(x,y)
=lim
l→
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
(+j)(+j)(lnx)
+j(lny)+j, for (x,y)∈(, ]×(, ],
ly+
(+j)(+j)(lnx)
+j|x>(lny)+j|y>
+ σ(y)ly
(+j)(+j)
[(ln)+j
+ [lnx+ (+j)ln](lnx)+j|x>
– (lnx)+j|x>](lny)+j|y>, for (x,y)∈(, ]×(, ]
()
⇒ u(x,y) =
⎧ ⎨ ⎩
(+j)(+j)
(lnx)+j(lny)+j, for (x,y)∈(, ]×(, ],
(+j)(+j)(lnx)
+j|x>(lny)+j|y>, for (x,y)∈(, ]×(, ].
By Lemma ., equation () is equivalent to system (). Therefore, equations ()-() satisfy the corresponding condition () of system (). Thus, equations ()-() satisfy all conditions of system (), that is, equations ()-() is the general solution of sys-tem ().
Appendix
In the section, we will prove the following conclusion.
(q)(q)H
J–q
(a+,b+)δxδy x
xi
lnx
s
q– y
b
σ(y)Ii
ux–i,ylny
t q–
h(s,t)dt
t
ds s
–
x
a
lnx
s
q– y
b
σ(y)Iiux–i,ylny
t q–
h(s,t)dt
t
ds s
= . (A.)
Proof Letσ(y)Ii(u(x–
i,y)) =ϑ(y). For the sake of convenience, we divide the calculation
into several steps.
Step. Compute
(q)(q)
HJ(–a+,qb+)δxδy x
xi
lnx
s
q– y
b
ϑ(y)
lny
t q–
h(s,t)dt
t
ds s
First of all,xx
+
By Definition ., we have
=
Substitute (A.)-(A.) into equation (A.), we have
Step. Compute
Therefore, by (A.) and (A.), we get
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The work described in this paper is financially supported by the National Natural Science Foundation of China (Grants Nos. 21576033, 21636004).
Received: 14 April 2016 Accepted: 20 October 2016
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